Given a set of J line correspondences, we can set up a system of linear equations similar to those for
points that involve matrix H and vector V, where V is as defined before and H is defined as follows
A a B a
C a A b
B b C b
A c B c
C c
1 1
1 1
1 1
1 1
1 1
1 1
1 1 1 1
1 1
P a P a
P a P b
P b P b
P b P c
P c a
b c
x 1
y 1
z 1
x 1
y 1
z 1
x 1
y 1
z 1
1 1
1
1 1
1 1
1 1
1 1
1
.
2 J=12
. H
s 12
Ž .
. A a
B a C a
A b B b
C b A c
B c C c
J J
J J
J J
J J
J J
J J
J J
J J
J J
P a P a
P a P b
P b P b
P c P c
P c a
b c
x J
y J
z J
x J
y J
z J
x J
Y J
z J
J J
J
J J
J J
J J
J J
J
and is called the coplanarity matrix. Again we can solve for V by minimizing the sum of squared
5 5
2
residual errors HV . V can be solved for up to a
scale factor. The scale factor can be determined by imposing one of the normality constraints as dis-
cussed for the case using points.
5. Camera transformation matrix from ellipse– circle correspondences
5.1. Camera transformation matrix from circles Ž
. Given the image an ellipse of a 3D circle and its
Ž .
size, the pose position and orientation of the 3D circle relative to the camera frame can be solved for
analytically. Solutions to this problem may be found Ž
. Ž
. in Haralick and Shapiro 1993 , Forsyth et al. 1991 ,
Ž .
and Dhome et al. 1989 . If we are also given the pose of the circle in the object frame, then we can
use the two poses to solve for R and T. Specifically,
Ž .
t
Ž .
t
let N s N N
N and O s O O O
be
c c
c c
c c
c c
x y
z x
y z
the 3D circle normal and center in the camera coor- Ž
dinate frame respectively. Also, let N s N N
o o
o
x y
.
t
Ž .
t
N and O s O
O O
be the normal and
o o
o o
o
z x
y z
center of the same circle, but in the object coordinate system. O and N are computed from the observed
c c
image ellipse using a technique described in Forsyth Ž
. et al. 1991 , while N and O are assumed to be
o o
known. The problem is to determine R and T from the correspondence between N and N , and be-
c o
tween O and O . The two normals and the two
c o
centers are related by the transformation R and T as shown below
N
o
x
r r
r
11 12
13
N
N sR N s 13
Ž .
o
r r
r
y
c o
21 22
23
r r
r
31 32
33
N
o
z
and O
t
o
x
r r
r
x 11
12 13
O t
O sRO qTs q
o
r r
r
y
y
c o
21 22
23
r r
r t
31 32
33
O
z o
z
14
Ž .
Ž .
Ž .
Equivalently, we can rewrite Eqs. 13 and 14 as follows
N r qN r qN r sN
o 11
o 12
o 13
c
x y
z x
N r qN r qN r sN
o 21
o 22
o 23
c
x y
z y
N r qN r qN r sN
o 31
o 32
o 33
c
x y
z x
and O r qO r qO r qt sO
o 11
o 12
o 13
x c
x y
x x
O r qO r qO r qt sO
o 21
o 22
o 23
y c
x y
x y
O r qO r qO r qt sO
o 31
o 32
o 33
z c
x y
x z
Each pair of 2D ellipse and 3D circle therefore offers six equations. The three equations from orientation
Ž Ž
.. Eq. 13
are not independent due to unity constraint on the normals. Given I observed ellipses and their
corresponding object space circles, we can set up a system of linear equations to solve for R and T by
5 5
2
minimizing the sum of residual errors QV yk ,
where Q and k are defined as follows
° ¶
N N
N
1 1
1
o o
o x
y z
N N
N
1 1
1
o o
o x
y z
N N
N
1 1
1
o o
o x
y z
O O
O 1
1 1
1
o o
o x
y z
O O
O 1
1 1
1
o o
o x
y z
O O
O 1
1 1
1
o o
o x
y z
.
6 I=12
. Q
s 15
Ž .
. N
N N
I I
I
o o
o x
y z
N N
N
I I
I
o o
o x
y z
N N
N
I I
I
o o
o z
y z
O O
O 1
I I
I
o o
o x
y z
O O
O 1
I I
I
o o
o x
y z
O O
O 1
¢ ß
I I
I
o o
o x
y z
and
t 6 I=1
N N
N O
O O
. . . N N
N O
O O
1 1
1 1
1 1
I I
I I
I I
k s
16
Ž .
c c
c c
c c
c c
c c
c c
x y
z x
y z
x y
z x
y z
ž
Since each circle provides six equations, a minimum Ž
. of two circles are needed if only circles are used to
uniquely solve for the 12 parameters in the transfor- mation matrix. To retain a linear solution, not even
one normality constraint can be imposed using La- grange multipliers due to k being non-zero vector.
6. The integrated technique
